CRN 12109: HW 5

Problem 21. Exercise 3.2.2 (a-d)

Problem 22. Given a set $X$ of real numbers, let $L$ be the set of all limit points of $X$. Show that $L$ is closed.

Problem 23. Show: If $X$ is both open and closed, then $X={\mathbb R}$ or $X=\emptyset$.

Problem 24. Exercise 3.2.14

Problem 25. A set $X$ is called limit point compact, if every infinite subset $A$ of $X$ has a limit point belonging to $A$.

Show that a set is limit point compact if and only if it is closed and bounded.